This protocol targets the limiting case of extremely rare events, while generically, we do not expect to gain an advantage from logical sensing at large signal rates.

Similar approaches have recently been explored to stochastically implementing small-angle transversal logical gates in quantum codes.

The precise requirements on the size of $\epsilon$ will be clarified below.

The sensitivity changes for $\epsilon \sim 1/n$, since, in that regime, sensing intervals with and without signal pulse can be distinguished with single shots with high success probability.

In fact, it is not necessary to perform the error correction step; performing the syndrome measurements suffices.

Of course, in practice, there is a lower limit for $\mathrm{\Gamma }$ set by the error-correction cycle time $\delta t$.

In Figs. 4–4, we evaluate Eq. (7) throughout. For small $\epsilon$, the optimal sensing interval $1/(2\mathrm{\Gamma })$ can exceed the total sensing time $T$, such that a single sensing interval $\mathrm{\Delta }{t}_L=T$ is optimal [regime (i)] and the achievable Fisher information is smaller. We neglect this effect here.

Note that, in both cases, we are working in the many-pulse regime, where the number of pulses per logical sensing interval is large, i.e., $RT\gg 1$. Especially in the second case, (ii), this number can become so large that the average acquired phase ${\mathrm{\Phi }}_L\sim 1/\epsilon$ exceeds $2\pi$, which needs to be taken into account appropriately.

Such noise processes can also be addressed with decoherence-free subspaces.